| # Binary Indexed Tree | |
| ## Problem Description | |
| Maintain prefix sums with point updates using a binary indexed tree. | |
| ## Input | |
| The functions receive the array representation, its logical size or capacity, and the operation-specific values described by their C signatures. | |
| ## Output | |
| Each operation returns or updates the represented data structure according to its interface contract. | |
| ## Constraints | |
| All indices, sizes, capacities, and stored values satisfy the bounds stated by the corresponding function contract. | |
| ## Source | |
| Classical data-structure implementation reconstructed from the verified C artifact. | |